Highly oscillatory solutions of a Neumann problem for a $p$-laplacian equation

Abstract

We deal with a boundary value problem of the form $-\epsilon ({\varphi }_p(\epsilon u^{\prime }){)}^{\prime }+a(x)W^{\prime }(u)=0,u^{\prime }(0)=0=u^{\prime }(1),$ where ${\varphi }_p(s)=|s{|}^{p-2}s$ for $s\in \mathbb{R}$ and $p>1$, and $W:[-1,1]\to \mathbb{R}$ is a double-well potential. We study the limit profile of solutions when $\epsilon \to 0^{+}$ and, conversely, we prove the existence of nodal solutions associated with any admissible limit profile when $\epsilon$ is small enough.